Which among the following functions is not injective
A
f(x)=|x+1|,x∈[−1,∞)
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B
g(x)=x+1x,x∈(0,∞)
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C
h(x)=x2+4x−5,x∈(0,∞)
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D
k(x)=e−x,x∈[0,∞)
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Solution
The correct option is Bg(x)=x+1x,x∈(0,∞) f(x)=|x+1|,x∈[−1,∞)
h(x)=x2+4x−5,x∈(0,∞) k(x)=e−x,x∈[0,∞)
Clearly, from above graphs f(x),h(x),k(x) using horizontal line test in the given intervals of each function, we can say that
they are injective functions in the given interval.
For g(x)=x+1x,x∈(0,∞) g′(x)=1−1x2=x2−1x2
Clearly, g′(x) is not continuously positive or continuously negative. Hence g(x) is increasing for some interval ((−∞,1)∪(1,∞))and decreasing for some interval ([−1,0)∪(0,1]) . So it is a many-one function.